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Mirrors > Home > ILE Home > Th. List > elequ1 | GIF version |
Description: An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
elequ1 | ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-13 1404 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | |
2 | ax-13 1404 | . . 3 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧)) | |
3 | 2 | equcoms 1594 | . 2 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧)) |
4 | 1, 3 | impbid 120 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-gen 1338 ax-ie2 1383 ax-8 1395 ax-13 1404 ax-17 1419 ax-i9 1423 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: cleljust 1813 elsb3 1852 dveel1 1896 nalset 3887 zfpow 3928 mss 3962 zfun 4171 bj-nalset 10015 bj-nnelirr 10078 |
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